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Objectives:
1.
Understand what a vector valued function is.
2.
Be able to do calculus with vector valued functions: differentiation, integration.
3.
Understand what the derivative of a vector valued function represents geometrically.
4.
Be able to find the arc length of a curve between two points.

Recap Video

You can watch watch the following video which recaps the ideas of the section.

Test your understanding with the following questions.

Parametrization Examples

Below are three videos showing examples of parametrizing curves.

Problems

Parametrize the intersection of the parabolic sheet with the plane .
Parametrize the curve in 3D given by in the plane .
Suppose is the curve given by .
  • The vector .
  • The derivative .
  • A tangent vector to at the point is . Plugging this into the derivative gives the vector

Suppose . If , then
Suppose a particle is moving with position function . At what value of is the speed of the particle smallest?

Find the arc length of the curve between the points and .

The arc length of the curve between the points and is: .